Problem 6 - Basic Statistics for Business & Economics

Question

The director of admissions at Kinzua University in Nova Scotia estimated the distribution of student admissions for the fall semester on the basis of past experience. What is the expected number of admissions for the fall semester? Compute the variance and the standard deviation of the number of admissions. $$ \begin{array}{|cc|} \hline \text { Admissions } & \text { Probability } \\ \hline 1,000 & .6 \\ 1,200 & .3 \\ 1,500 & .1 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Expected admissions: 1110; Variance: 24900; Standard deviation: 157.76.

Step 1: Calculate expected value

To find the expected number of admissions, we multiply each number of admissions by its corresponding probability and sum these products. The formula for expected value \( E(X) \) is given by \( E(X) = \sum_{i} x_i \cdot p_i \). Substituting in the given values: \( E(X) = 1000 \times 0.6 + 1200 \times 0.3 + 1500 \times 0.1 \). Calculating this gives us: \( E(X) = 600 + 360 + 150 = 1110 \). Hence, the expected number of admissions is 1110.

Step 2: Compute variance

To find the variance, we calculate \( Var(X) = \sum_{i} (x_i - \mu)^2 \cdot p_i \), where \( \mu \) is the expected value from Step 1. Therefore, \( Var(X) = (1000-1110)^2 \times 0.6 + (1200-1110)^2 \times 0.3 + (1500-1110)^2 \times 0.1 \). Performing the calculations gives \( (1000-1110)^2 = 12100 \), \( (1200-1110)^2 = 8100 \), \( (1500-1110)^2 = 152100 \). So, \( Var(X) = 12100 \times 0.6 + 8100 \times 0.3 + 152100 \times 0.1 = 7260 + 2430 + 15210 = 24900 \). Thus, the variance is 24900.

Step 3: Calculate standard deviation

The standard deviation is the square root of the variance, \( \sigma = \sqrt{Var(X)} \). From Step 2, \( Var(X) = 24900 \). Thus, \( \sigma = \sqrt{24900} \). The calculation gives \( \sigma \approx 157.76 \). Therefore, the standard deviation of the number of admissions is approximately 157.76.

Key Concepts

variance

Variance is a key concept in statistics that helps us measure how much a set of numbers differs, or varies, from the average. This measure is particularly important in probability distribution, where it tells us about the spread of the possible outcomes. In our exercise, we came across the variance as part of calculating how much the student admissions at Kinzua University deviate from the expected number of 1110. But what exactly is variance and how is it calculated? To find variance, you need to follow these simple steps:

• First, determine the expected value, which in this case is 1110. It represents the mean of all possible outcomes based on their probabilities.
• Next, subtract this expected value from each outcome.
• Square the result for each outcome to eliminate negative numbers and give more weight to larger differences.
• Finally, multiply each squared result by its corresponding probability, and sum these products to get the variance 24900.

Variance provides a numerical value that informs us how much the admissions numbers spread out from the average, with a higher variance indicating a wider spread.

standard deviation

The standard deviation is closely related to variance and is another important statistical measure used to describe the variability or spread of a data set. While variance gives us a sense of how data points differ from the mean, standard deviation puts this into the context of the original data units, making it easier to interpret. To compute the standard deviation, you simply take the square root of the variance. In our Kinzua University example, this was computed as follows:

• The variance was calculated as 24900.
• By taking the square root of 24900, we get a standard deviation of approximately 157.76.

When interpreting this, a standard deviation of 157.76 means that individual admission numbers deviate from the mean of 1110 by about 157.76 admissions on average. The standard deviation provides a straightforward way to grasp how spread out the admission numbers are against the expected mean.

probability distribution

Probability distribution is a fundamental concept in statistics. It describes how probabilities are assigned to different possible outcomes of a random variable. In essence, a probability distribution tells us the chance or likelihood of each outcome occurring. In our exercise involving Kinzua University's admissions, we have the probability distribution as follows:

• There is a 60% probability (0.6) that admissions will be 1000.
• There is a 30% probability (0.3) that admissions will be 1200.
• And finally, there is a 10% probability (0.1) that admissions will be 1500.

This distribution gives us a complete view of all possible admission numbers and how likely each is to occur. By understanding the distribution, we are able to compute expected values, as well as variance and standard deviation. Probability distributions are essential in making informed predictions and assessing the risk or variability of outcomes in different scenarios.